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Misalignment and Resonance Torques and Their Treatment in GP-B Data Analysis

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### Misalignment and Resonance Torques and Their Treatment in GP-B Data Analysis

Mac Keiser and Alex Silbergleit

Outline

- Misalignment Torques
- Observations
- Explanation and Calculation of Torque
- Data Analysis
- Resonance Torques
- Observations
- Explanation and Calculation of Torque
- Data Analysis
- Summary

Gravity Probe B Mission Timeline

Initialization

Phase

Science Data Collection

Phase

Calibration

Phase

Launch

April 20, 2004

Gyroscopes Spun Up and Aligned

August 29, 2004

Aug. 15, 2005

Liquid Helium Depeleted

Sept. 29, 2005

Proton Flux, Jan. 20-22, 2005,

Measured by GOES Satellite

Gyro 3 West-East Spin Axis Orientation

Particles/(cm2 sec sr)

Arc Sec

Time (days) from Jan. 1, 2005

Time (days)

Misalignment Torque - ObservationsGravity Probe B Mission Timeline

NhS1

(acquired)

Initialization

Phase

Science Data Collection

Phase

Calibration

Phase

HR Peg

(acquired)

Launch

April 20, 2004

Gyroscopes Spun Up and Aligned

August 29, 2004

Aug. 15, 2005

Liquid Helium Depleted

Sept. 29, 2005

20

- Calibration Phase Spacecraft Maneuvers
- Increased the Misalignment Between the Satellite Roll Axis and the Gyroscope Spin Axes
- 19 Maneuvers to Nearby Stars or “Virtual” Stars
- Operating Conditions Changed
- DC or AC Suspension Voltages
- Spacecraft Attitude Control

IM Peg

Guide Star

20

Calibration Phase ObservationsMisalignment TorquesObservations – Gyroscope 3

Gyroscope 3, Mean Rate (mas/day) vs. Mean Misalignment (as)

Mean North-South Misalignment

Mean West-East Misalignment

4000

120

60

90

4000

120

60

3000

3000

150

30

2000

2000

150

30

1000

1000

180

0

180

0

210

330

210

330

240

300

240

300

270

270

90

4000

120

90

3000

120

60

2000

150

30

150

30

1000

180

0

180

0

210

330

210

330

240

300

300

240

270

270

Observations – All GyroscopesGyroscope 1

Gyroscope 2

Mean North-South Misalignment

Gyroscope 4

Gyroscope 3

4000

Mean North-South Misalignment

Mean West-East Misalignment

Mean West-East Misalignment

Observations–Change of Electrode PotentialGyroscope Drift Rates, DC Preload, Misalignment 10

Dipole

Layer

Non-uniform potential

Calculation of Torque due to Patch Effect FieldsElectric Field at a Metallic Surface

E

Uniform Potential

No Patch Effect Field

Torques due to Patch Effect Potential on Rotor and Housing

- Expand Potential on Each Surface in Terms of Spherical Harmonics

- Use Rotation Matrices to Transform to a Common Reference Frame

- Solve Laplace’s equation, find energy stored in electric field

- Find the torque by differentiating the energy with respect to the angles which determine the mutual orientation of the conductors

Analytical Expression for Torque

Torque Coefficient

- Proportional to Misalignment

- Perpendicular to Misalignment Direction

- Modulated at Polhode Frequency

- Depends of Polhode Path

- Depends of Patch Effect on Rotor and Housing

Torque

roll

spin

Misalignment Torques - Data Analysis

Is it possible to separate the gyroscope drift rate due to misalignment torques from the drift rate due to relativistic effects?

Characteristics of Misalignment and Uniform Drift

Simulated Data

- Radial Component of Drift Rate Contains NO Contribution from Misalignment Drift
- Magnitude and Direction of Uniform (Relativistic) Drift Rate May Be Determined From Variation of Radial Component with Misalignment Phase

Two Data Analysis Methods

- Explicitly Include Misalignment Torques in Analysis of Data
- Only Use Information on Radial Rate
- Precision of Drift Rate Estimates ~ 1/T3/2
- Initial Application of This Method In N Batches ~ N/T3/2
- New Data Analysis Approach Recovers Full Precision
- Explicit Use of Sequential Correlated Noise in Rate Estimates

Resonance Torques

Observation*: Offsets in Orientation of Gyroscope Axis Tend to Occur when a harmonic of the gyroscope polhode frequency is equal to the satellite roll frequency

Roll Frequency = 143 * Polhode Frequency

* J. Kolodziejczak, MSFC

Analytical Expression for Torque

Torque Components

Calculation of Patch Effect Resonance Torque:Harmonic of Polhode Frequency Equal to Roll FrequencyTorque

spin

roll

- Properties of Resonance Torques
- Resonance Condition, nfp = fr
- Independent of Misalignment
- Direction Depends on Relative Phase and Distribution of Patches
- Depends on Polhode Path

Resonance Torques – Predicted Cornu Spiral

Fresnel Integrals: Integration of Equations of Motion With Linearly Varying Polhode Frequency, Constant Polhode Angle

Resonance Torques: Data Analysis

- Exclude data in vicinity of resonances
- Explicitly include resonances in data analysis
- Two Parameters Uniquely determine each resonance

Example: Analysis of Data for Gyroscope 4

Misalignment Torques: Use only radial rate information (along the misalignment vector)

Resonance Torques: Exclude Data in Vicinity of Resonance

Formal Statistical Rate Errors:

NS = 16 mas/yr

WE = 14 mas/yr

Summary

- Patch Effect Torques are dominant classical torques acting on the gyroscopes
- Motion of gyroscope spin axis due to patch effect torques can be separated from the relativistic motion of the gyroscopes.
- Misalignment Torque:
- Acts in Direction Perpendicular to Misalignment
- Resonance Torque
- Displacement Occurs in Finite Time
- Unique Time Signature

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